It is known that a passive contaminant, introduced into a pipe or open channel of constant cross-section, will disperse in the direction of flow so that the distribution of concentration, averaged over the cross-section, is eventually a Gaussian function of distance along the pipe axis. It is argued that the effect of the viscous sub-layer, in which the fluid velocity is small and in which lateral mixing is of low intensity, is often such that there is an intermediate stage in the longitudinal dispersion process in which that part of the profile of the distribution of concentration which arises from contaminant outside the sub-layer is itself Gaussian, but with the value of the longitudinal diffusivity determined by conditions outside the sub-layer. This intermediate stage evolves very gradually into the final stage in which the value of longitudinal diffusivity is determined by conditions throughout the whole cross-section. A model is described to illustrate this process and the existence of the intermediate stage is confirmed by calculation. A product of the calculation is the exact value of the covariance of the longitudinal velocities of a fluid molecule at times separated by an interval t, and the paper concludes with a discussion of the properties of the calculated covariance, and its relevance to real flows.